The Power-law Formalism as a Tool for Modeling Hormonal Systems

Abstract

Modeling a hormone system requires a number of simplifying assumptions. Often, the final conceptual model incorporates a number of aggregated processes that have no correspondence with a single enzyme-catalyzed reaction. In such cases, it is discussible using models based on classical biochemical kinetics rate laws that are valid only under specific conditions. The power-law formalism provides an alternative framework for building up a mathematical model in such cases. The resulting model is a set of ordinary differential equations with a special structure that allows efficient symbolic and numerical analysis of the system's properties. In these equations, the underlying rate-laws of each of the component processes are represented by a power-law that is an exact representation of the actual rate-law at the operating point. The particular form of these equations allows representation of a wide range of kinetic features without changing the basic power-law form. Moreover, its parameters have an immediate interpretation as apparent kinetic-orders and rate-constants. This is especially helpful for incorporating both quantitative and qualitative information in the process of model definition. This is particularly useful when detailed kinetic information concerning system's components is not available. In this paper we show the utility of the power-law approach in this context by deriving an illustrative model of a complex physiological system: the hypothalamus-anterior pituitary-thyroid network. First, we derive a conceptual model that incorporates the key features of this system. Then, we derive an S-system model, one of the preferred variants within the power-law formalism, and we show its utility in exploring the system properties. The model qualitatively reproduces the response of normal, hyperthyroid, and hypothyroid patients to a clinical test involving a thyrotropin releasing hormone injection. Finally, we illustrate the utility of this modeling strategy for studying the system's response to different dynamic patterns of regulatory signals, and for exploring how altered dynamic patterns of stimulatory signals can cause pathological states.